CoFiGAN: Collaborative Filtering by Generative and Discriminative - - PowerPoint PPT Presentation

CoFiGAN: Collaborative Filtering by Generative and Discriminative Training for One-Class Recommendation

Jixiong Liua,b,c, Weike Pana,b,c∗ and Zhong Minga,b,c∗

liujixiong@email.szu.edu.cn, {panweike, mingz}@szu.edu.cn aNational Engineering Laboratory for Big Data System Computing Technology

Shenzhen University, Shenzhen, China

bGuangdong Laboratory of Artificial Intelligence and Digital Economy (SZ)

Shenzhen University, Shenzhen, China

cCollege of Computer Science and Software Engineering

Shenzhen University, Shenzhen, China

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Introduction

Problem Definition

One-Class Collaborative Filtering (OCCF) Input: Observations in the form of (user, item) pairs. Training: A min-max game between a generator and a discriminator, in which the generator improves the quality of generated items to fool the discriminator. Goal: For each target user u, we provide a personalized list of items from I\Iu via the generator, where I and Iu denote the set

• f all items and the set of user u’s observed items, respectively.

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Introduction

Challenges

Non-differentiable: The discrete sampling step prevents the loss from being back-propagated to the generator. Mode collapse: The generator becomes lazy to explore different items so as to avoid penalty from the discriminator.

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Introduction

Notations (1/3)

n number of users m number of items d number of latent dimensions bs batch size of mini-batch gradient descent Iu items preferred by u in the training data T1, T2, T3 numbers of iterations yui ∈ {0, 1} indicator whether (u, i) pair is observed U = {1, 2, . . . , n} whole set of users I = {1, 2, . . . , m} whole set of items (U, I)bs

• ne batch (u, i) pairs

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Introduction

Notations (2/3)

D discriminator UD

u· ∈ R1×d

latent feature vector of user u in D V D

i· ∈ R1×d

latent feature vector of item i in D bD

i ∈ R

bias of item i in D γD learning rate in D ID+

u

positive items generated by D ID−

u

negative items generated by D ˆ rD

ui = UD u·V D i· T + bD i

preference of user u to item i in D αD

u , αD v , βD v

tradeoff parameters in D

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Introduction

Notations (3/3)

G generator UG

u· ∈ R1×d

latent feature vector of user u in G V G

i· ∈ R1×d

latent feature vector of item i in G bG

i ∈ R

bias of item i in G γG learning rate in G IG−

u

negative items generated by G ˆ rG

ui = UG u·V G i· T + bG i

preference of user u to item i in G αG

u , αG v , βG v

tradeoff parameters in G

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Related Work

BPR-MF

In BPR-MF [Rendle et al., 2009], the users’ preference behaviors are modeled based on a pairwise preference assumption, i.e., a user u prefers an observed item i to an unobserved one j, Pre(u, i|φ) > Pre(u, j|φ), (1) where Pre(u, i|φ) and Pre(u, j|φ) denote user u’s preference to the

• bserved item i and the unobserved item j, respectively. And φ are the

model parameters that we would like to learn.

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Related Work

Logistic-MF

Logistic-MF [Johnson, 2014] assigns different labels to the observed items and the unobserved ones for each user, which is a pointwise algorithm, min

φ

• (u,i)∈P∪A

log(1 + exp(−ruiˆ rui)) + λ||φ||2, (2) where rui is the label of (u, i). And rui = 1, if (u, i) ∈ P; rui = 0,

• therwise. ˆ

rui is the predicted preference of user u to item i, and P, A represent the set of users’ observed items and the set of sampled negative ones in the training data, respectively.

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Related Work

IRGAN (1/3)

JG∗,D∗ = min

θ

max

φ n

• u=1

Ei∼ptrue log D(i|u) + Ei∼pθ(i|u) log(1 − D(i|u)), (3) where θ and φ denote the parameters of the generator and the discriminator respectively, ptrue denotes the distribution of users’ true preferences, pθ(i|u) denotes the item sampling probability given user u, and D(i|u) = σ(ˆ rD

ui ) denotes the estimated probability of item i

belonging to the ground truth data of user u.

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Related Work

IRGAN (2/3)

Discriminative training: The discriminator aims to discriminate the ground truth preference items from the items generated by the generator. Generative training: The generator aims to generate/sample high-quality items that look like the ground truth items for higher reward from the discriminator, and the generator update itself by the reward or penalty signal from the discriminator, i.e., policy gradient approximation [Sutton et al., 1999].

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Related Work

IRGAN (3/3) - Policy Gradient

∇θJG = ∇θEi∼pθ(i|u)[log(1 + exp(ˆ rD

ui )))]

=

m

• i=1

∇θpθ(i|u) log(1 + exp(ˆ rD

ui)))

=

m

• i=1

pθ(i|u)∇θ log pθ(i|u) log(1 + exp(ˆ rD

ui ))

= Ei∼pθ(i|u)[∇θ log pθ(i|u) log(1 + exp(ˆ rD

ui ))]

≈ 1 K

K

• i=1

∇θ log pθ(i|u) log(1 + exp(ˆ rD

ui )),

(4) where pθ(i|u) denotes the sampling probability of item i via the generator, K is the number of generated samples, and log(1 + exp(ˆ rD

ui )) is the reward/penalty from the discriminator.

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Our Method

Our Solution: CoFiGAN

Our CoFiGAN also plays a min-max game inheriting from GAN, JG∗,D∗ = min

θ

max

φ n

• u=1
• Ei∼ptrue log D(i|u) + Ei∼pθ(i|u) log(1 − D(i|u))
• ,

(5) where φ and θ are parameters of the discriminator and generator, respectively, and D(i|u) denotes the estimated probability of item i belonging to the ground truth data of user u. Notice that we propose a more direct approximation of the generator’s loss function, which yields the main difference compared with IRGAN.

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Our Method

Discriminative Training (1/2)

φ∗ = arg max

φ n

• u=1

[Ei∼ptrue log D(i|u) + Ei∼pθ(i|u) log(1 − D(i|u))]. (6) The discriminator aims to label the ground truth data and the generated samples as ‘1’ and ‘0’ separately, thus it is natural to take the binomial Logistic regression model as a discriminator. min

φ

• (u,i)∈(U,I)bs

f D

ui ,

(7) where f D

ui = (1 − yui) · ˆ

rD

ui + log(1 + exp−ˆ r D

ui ) + αD u

2 UD u·2 + αD

v

2 V D i· 2 + βD

v

2 bD i 2

where label yui = 1 if (u, i) is a ground truth pair and yui = 0 otherwise.

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Our Method

Discriminative Training (2/2)

In order to avoid numerical overflow of exp(−ˆ rD

ui ) when ˆ

rD

ui < 0, we

follow the same trick used in TensorFlow1, and use the same following

• bjective functions in our own implementation using the Java

programming language. Sine e is used as the base of the log function in our experiments, we can easily verify that the following function is equivalent to f D

ui above.

f D

ui =

       (1−yui)ˆ rD

ui + log(1+e−ˆ r D

ui )+ λD

2 φ2, ifˆ rD

ui ≥ 0

−ˆ rD

ui · yui +log(1+eˆ r D

ui )+ λD

2 φ2, if ˆ rD

ui < 0

, and λD

2 φ2 = αD

u

2 UD u·2+ αD

v

2 V D i· 2+ βD

v

2 bD i 2.

1Implementation of sigmoid cross entropy with logits in TensorFlow Liu, Pan and Ming (SZU) CoFiGAN 14 / 40

Our Method

Generative Training (1/3)

G∗ = arg min

θ n

• u=1

Ei∼pθ(i|u) log(1 − D(i|u)) = arg max

θ n

• u=1

Ei∼pθ(i|u)[log(1 + exp(ˆ rD

ui ))],

(8) where pθ(i|u) denotes the item sampling probability given user u calculated by the generator, and log(1 + exp(ˆ rD

ui ) can be interpreted as

the reward/penalty scored by the discriminator. And thus the update rule of the generator can be interpreted as encouraging sampling items with high reward in the next sampling step.

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Our Method

Generative Training (2/3)

To solve Eq.(8), we propose a more direct approximation by narrowing the distance between the samples generated by the generator and the discriminator, which contrasts with policy gradient approximation, θ∗ = arg min

θ n

• u=1

ESG∼pθ(S|u)[Dist(SG, SD)], (9) where SG and SD are high-quality samples that can confuse the discriminator from the perspectives of the generator and discriminator, respectively.

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Our Method

Generative Training (3/3)

To narrow Dist(SG, SD) in Eq.(9), the generator is designed to generate items close to high-quality ones and away from low-quality

• nes from the perspective of the discriminator, i.e., ID+

u

and ID−

u

, min

θ

• u∈U
• i∈ID+

u

• j∈ID−

u

f G

uij,

(10) where θ = {UG

u·, V G i· , bG i , u ∈ U, i ∈ I}, σ is the sigmoid function, and

f G

uij = 1 |Iu|(− log(σ(ˆ

rG

ui − ˆ

rG

uj )) + αG

u

2 UG u·2 + αG

v

2 V G i· 2 + αG

v

2 V G j· 2 + βG

v

2 bG i 2 + βG

v

2 bG j 2), which is the same as BPR-MF.

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Our Method

Relationship with IRGAN (1/3)

• Figure: Illustration of IRGAN and our CoFiGAN with a discriminator D and a

generator G. In both IRGAN and our CoFiGAN, for a user u, orange balls (Iu) and green balls (IG−

u

) denote true positive samples and generated fake samples to be exploited in discriminative training. In IRGAN, G aims to generate items (purple balls) associated with more reward (r) rather than penalty(p) in which the signals are from D. In our CoFiGAN, for a user u, G tries to generate items close to ID+

u

(red balls) instead of ID−

u

(blue balls) in a pairwise way, which is a more direct and effective method to influce the next generation round because D in our CoFiGAN provides the direct guidance (ID+

u

and ID−

u

), while D in IRGAN gives the indirect guidance (r/p).

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Our Method

Relationship with IRGAN (2/3) - Similarities

The discriminator in IRAGN and CoFiGAN are designed the same as a binary classifier, which aims to discriminate the ground truth data from the generated samples accurately. The generator and discriminator in IRGAN and CoFiGAN are factorization-based models, i.e., users’ preferences are estimated by the inner product of a user’s and an item’s latent feature vectors. The time complexity in IRGAN and our CoFiGAN are O(T1nK log M), which highly depends on the number of iterations T1, the number of users n, the number of generated samples K, and the size of candidate items M.

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Our Method

Relationship with IRGAN (3/3) - Differences

The main difference between IRGAN and our CoFiGAN is the design

• f the approximation for the generative training.

In IRGAN, the generator generates samples that will get corresponding reward or penalty from the discriminator, which encourages generating higher-reward samples in the next sampling round. In CoFiGAN, a more direct approximation is designed. Owing to the direct guidance ID+

u

and ID−

u

, the generator can effectively expand the margin between these two sets of items.

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Our Method

Advantages

Higher efficiency: With the positive and negative samples provided by the discriminator, our generator generates items close to high-quality items and away from low-quality ones in a pairwise way. Larger capacity: With the positive and negative samples provided by the discriminator, the generator is encouraged to explore different samples in different sampling rounds, which may avoid mode collapse to some extent.

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Our Method

Gradients

∂f G

uij

∂θ = ∂f G

uij

∂rG

uij

× ∂rG

uij

∂θ + λGθ = 1 |Iu| × [−σ(−ˆ rG

ui + ˆ

rG

uj ) ×

∂rG

uij

∂θ + λGθ], (11) where λG is the tradeoff parameter and it can be αG

u , αG v or βG v for the

corresponding model parameter. ∂f D

ui

∂φ =          [1 − yui − σ(−ˆ rD

ui)] × ∂ˆ

rD

ui

∂φ + λDφ, if ˆ rD

ui ≥ 0

[−yui + σ(ˆ rD

ui )] × ∂ˆ

rD

ui

∂φ + λDφ, if ˆ rD

ui < 0

, (12) where λD is the tradeoff parameter and can be αD

u , αD v or βD v according

to the corresponding model parameter.

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Our Method

Update Rules

θ = θ − γG × (

• (i,j),i∈ID+

u

,j∈ID−

u

∂f G

uij/∂θ),

(13) where θ can be UG

u·, V G i· or bG i , and γG is the learning rate.

φ = φ − γD ×

• (u,i)∈(U,I)bs

∂f D

ui /∂φ,

(14) where (U, I)bs is a set of (user, item) pairs in one bs (i.e., batch size) picking up, φ can be UD

u·, V D i· or bD i , and γD is the learning rate.

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Our Method

Algorithm

1: Pre-train the generator G and the discriminator D. 2: for t1 = 1, . . . , T1 do 3:

for t2 = 1, . . . , T2 do

4:

Discriminative training.

5:

end for

6:

for t3 = 1, . . . , T3 do

7:

Generative training.

8:

end for

9: end for

Figure: The algorithm of our CoFiGAN.

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Our Method

Algorithm

1: if t2%k = 0 then 2:

for u ∈ U (notes: in ascending order w.r.t. user ID) do

3:

Generate IG−

u

.

4:

end for

5: end if 6: for t3 = 1, . . . ,

u∈U 2|Iu|/bs (notes: in ascending order w.r.t. user

ID) do

7:

Construct (U, I)bs by alternatively taking a positive and negative item.

8:

Calculate the gradients of the model parameters in D via Eq.(12).

9:

Update the model parameters in D via Eq.(14).

10: end for

Figure: Discriminative training.

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Our Method

Algorithm

1: for u ∈ U (notes: in ascending order w.r.t. user ID) do 2:

Generate ID+

u

and ID−

u

via D.

3:

for (i, j), i ∈ ID+

u

, j ∈ ID−

u

do

4:

Accumulate the gradients of the model parameters in G via Eq.(11).

5:

end for

6:

Update the model parameters in G via Eq.(13).

7: end for

Figure: Generative training.

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Experiments

Datasets

Table: Statistics of the first copy of each of the four datasets used in the

• experiments. And |Itr

u |, |Iva u |, |Ite u | show the number of pairs in the training

data, validation data and test data, respectively.

Dataset |U| |I| |Itr

u |/|Iva u |/|Ite u |

ML100K 943 1682 26747/941/27687 ML1M 6040 3952 281605/6036/287640 UserTag 3000 2000 120234/2984/123218 NF5K5K 5000 5000 73320/4616/77936

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Experiments

Baselines

PopRank BPR-MF [Rendle et al., 2009] Logistic-MF [Johnson, 2014] Neural-MF [He et al., 2017] IRGAN [Wang et al., 2017]

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Experiments

Evaluation Metrics

Precision@k NDCG@k

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Experiments

Parameters Configurations (1/2)

For BPR-MF and Logistic-MF, we fix the maximum iteration number 1000, latent dimension d = 20 and learning rate γ = 0.01, and select the best tradeoff parameters and iteration number via Precision@5 on the validation data. We use BPR-MF as the pretrain generator for IRGAN and CoFiGAN, and Logistic-MF as the pretrain discriminator for CoFiGAN. For Neural-MF, we keep the same architecture and tune the parameters according to the original paper.

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Experiments

Parameters Configurations (2/2)

For IRGAN, we fix latent dimension d = 20, learning rate γG = γD = 0.001 because it does not perform well when using γG = γD = 0.01, and we select the best value of the regularization parameters λG and λD from {0.001, 0.01, 0.1}. Finally, we select “λG = 0.001, λD = 0.1”. For CoFiGAN, we fix latent dimension d = 20, learning rate γG = γD = 0.01, and we select the best value of the regularization parameters λG and λD from {0.001, 0.01, 0.1}. Finally, we select “λG = 0.001, λD = 0.01”.

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Experiments

Main Results (1/5)

Table: Precision@k of PopRank, BPR-MF, Logistic-MF, IRGAN, Neural-MF and our CoFiGAN on ML100K and ML1M. The best method and corresponding results are marked in bold.

Dataset Method Precision@3 Precision@5 Precision@10 ML100K PopRank 0.3044±0.0004 0.2732±0.0090 0.2356±0.0047 BPR-MF 0.3923±0.0097 0.3652±0.0080 0.3248±0.0039 Logistic-MF 0.4142±0.0236 0.3878±0.0193 0.3478±0.0129 Neural-MF 0.3893±0.0061 0.3689±0.0068 0.3247±0.0073 IRGAN 0.3915±0.0087 0.3665±0.0067 0.3253±0.0041 CoFiGAN 0.4333±0.0079 0.4053±0.0078 0.3573±0.0026 ML1M PopRank 0.3051±0.0010 0.2824±0.0018 0.2552±0.0012 BPR-MF 0.4418±0.0005 0.4201±0.0032 0.3811±0.0030 Logistic-MF 0.4681±0.0024 0.4432±0.0020 0.3980±0.0022 Neural-MF 0.4227±0.0040 0.3993±0.0031 0.3584±0.0025 IRGAN 0.3219±0.0131 0.3098±0.0120 0.2927±0.0114 CoFiGAN 0.4769±0.0026 0.4484±0.0037 0.4007±0.0048

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Experiments

Main Results (2/5)

Table: Precision@k of PopRank, BPR-MF, Logistic-MF, IRGAN, Neural-MF and our CoFiGAN on UserTag and NF5K5K. The best method and corresponding results are marked in bold.

Dataset Method Precision@3 Precision@5 Precision@10 UserTag PopRank 0.2869±0.0011 0.2683±0.0011 0.2385±0.0010 BPR-MF 0.3083±0.0054 0.2912±0.0029 0.2548±0.0008 Logistic-MF 0.3086±0.0073 0.2909±0.0091 0.2607±0.0064 Neural-MF 0.3047±0.0023 0.2901±0.0046 0.2573±0.0029 IRGAN 0.3073±0.0042 0.2919±0.0025 0.2565±0.0005 CoFiGAN 0.3227±0.0033 0.3045±0.0039 0.2672±0.0061 NF5K5K PopRank 0.1999±0.0022 0.1928±0.0020 0.1706±0.0037 BPR-MF 0.2624±0.0050 0.2477±0.0040 0.2214±0.0017 Logistic-MF 0.2503±0.0095 0.2343±0.0068 0.2075±0.0054 Neural-MF 0.2414±0.0040 0.2291±0.0010 0.2031±0.0013 IRGAN 0.2200±0.0016 0.2107±0.0017 0.1955±0.0004 CoFiGAN 0.2884±0.0053 0.2679±0.0051 0.2347±0.0033

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Experiments

Main Results (3/5)

Table: NDCG@k of PopRank, BPR-MF, Logistic-MF, IRGAN, Neural-MF and

• ur CoFiGAN on ML100K and ML1M. The best method and corresponding

results are marked in bold.

Dataset Method NDCG@3 NDCG@5 NDCG@10 ML100K PopRank 0.3175±0.0029 0.2927±0.0067 0.2653±0.0021 BPR-MF 0.4009±0.0121 0.3816±0.0110 0.3607±0.0050 Logistic-MF 0.4249±0.0240 0.4059±0.0206 0.3866±0.0155 Neural-MF 0.3973±0.0052 0.3831±0.0039 0.3609±0.0056 IRGAN 0.4002±0.0113 0.3825±0.0102 0.3612±0.0038 CoFiGAN 0.4425±0.0075 0.4228±0.0074 0.3972±0.0011 ML1M PopRank 0.3110±0.0004 0.2938±0.0010 0.2736±0.0008 BPR-MF 0.4488±0.0023 0.4323±0.0027 0.4052±0.0028 Logistic-MF 0.4773±0.0026 0.4581±0.0023 0.4268±0.0025 Neural-MF 0.4314±0.0055 0.4133±0.0042 0.3844±0.0030 IRGAN 0.3249±0.0141 0.3159±0.0131 0.3047±0.0126 CoFiGAN 0.4856±0.0022 0.4641±0.0029 0.4300±0.0045

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Experiments

Main Results (4/5)

Table: NDCG@k of PopRank, BPR-MF, Logistic-MF, IRGAN, Neural-MF and

• ur CoFiGAN on UserTag and NF5K5K. The best method and corresponding

results are marked in bold.

Dataset Method NDCG@3 NDCG@5 NDCG@10 UserTag PopRank 0.2899±0.0006 0.2766±0.0012 0.2548±0.0011 BPR-MF 0.3136±0.0051 0.3005±0.0036 0.2740±0.0018 Logistic-MF 0.3145±0.0066 0.3010±0.0078 0.2791±0.0060 Neural-MF 0.3085±0.0024 0.2977±0.0040 0.2744±0.0031 IRGAN 0.3127±0.0035 0.3007±0.0026 0.2750±0.0008 CoFiGAN 0.3272±0.0036 0.3138±0.0037 0.2867±0.0054 NF5K5K PopRank 0.2013±0.0005 0.1985±0.0011 0.1921±0.0025 BPR-MF 0.2703±0.0054 0.2640±0.0046 0.2608±0.0030 Logistic-MF 0.2591±0.0093 0.2538±0.0074 0.2533±0.0065 Neural-MF 0.2495±0.0044 0.2442±0.0017 0.2399±0.0004 IRGAN 0.2247±0.0023 0.2223±0.0012 0.2263±0.0008 CoFiGAN 0.2993±0.0050 0.2894±0.0052 0.2817±0.0049

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Experiments

Main Results (5/5)

25 50 75 100 125 150 175 200 epoch 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Prec@5 Generator Discriminator 25 50 75 100 125 150 175 200 epoch 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Prec@5 Generator Discriminator

ML100K ML1M

25 50 75 100 125 150 175 200 epoch 0.05 0.10 0.15 0.20 0.25 0.30 Prec@5 Generator Discriminator 25 50 75 100 125 150 175 200 epoch 0.05 0.10 0.15 0.20 0.25 Prec@5 Generator Discriminator

UserTag NF5K5K

Figure: Recommendation performance of the generator (G) and discriminator (D) with different epochs in our CoFiGAN.

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Experiments

Observations

Our CoFiGAN performs much better than all the baseline methods across the four datasets, which shows the effectiveness of the proposed adversarial training algorithm. BPR-MF and Logistic-MF are very competitive, which shows the effectiveness of the pairwise and pointwise preference assumptions for modeling the one-class feedback data. IRGAN does not perform very well, which may be caused by the fact that the generator in IRGAN falls into mode collapse. Our CoFiGAN achieves better performance consistently on all the four different datasets, while IRGAN performs well on UserTag

• nly, which shows the robustness of our CoFiGAN.

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Conclusions

Conclusions

We have studied a classical recommendation problem, i.e.,

• ne-class collaborative filtering (OCCF), which aims to provide a

ranked list of items for each user based on the implicit feedback. Our CoFiGAN unifies a generator and a discriminator, and plays a min-max game including generative training and discriminative training, which can reach an equilibrium when the discriminator can hardly distinguish the ground truth data from the samples generated by the generator.

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Thank you

Thank you!

We thank the Editor in Chief Prof. Jie Lu, the handling Associate Editor and the Reviewers for their efforts and constructive expert comments. We thank Miss Chuqian Zeng and Miss Shan Ouyang for their help on paper presentation. We thank the support of National Natural Science Foundation of China Nos. 61872249, 61836005 and 61672358.

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